The modeling of physical systems oftentimes requires the modeling of transport delays inherent in these physical systems. Models of physical systems that use transport delays include models of transportation systems, airports, or conveyor belt arrangements. These transport delays may be best viewed as the time required for a material to go through a buffer of know size at a known speed. The simplest forms of transport delays are exhibited by a constant delay assigned to a signal which results in the delivery of the input signal at a fixed time in the future. In situations where the speed of material transfer through the buffer is constant, a variable time delay may be computed.
An existing variable time delay is best exhibited by example. A common use of variable time delay is seen in the flow of an incompressible fluid through a pipe with an output located at length L distance away from the input. This length L can be best understood as a buffer between input and output. In such a setting, there will be a time delay td determined by the length of the transportation and the speed of the transportation. To simplify the example, it is assumed that the speed of the transportation (i.e. the fill rate of the buffer) at the inlet of the pipe is νi (t) and the speed at the outlet is νo(t). For further simplification, it will be assumed that νi (t) and νo (t) are equal, as the fluid traversing the length L is incompressible. Further associated with the present example are input ui and output uo which denote specific properties of the transported material, such as temperature at the inlet and outlet. Generally speaking, the relationship of ui to uo can be defined such that uo is a delayed ui and the delay is caused by the transportation. Representing this as an equation yields uo=ui(t−td).
To determine a variable time delay, it is necessary to solve for td based on the transport speed νi and the length L of the pipe. If the conservation variable at the inlet of the pipe is defined as
            L      i        ⁡          (      t      )        =            ∫              t        0            t        ⁢                            v          i                ⁡                  (          τ          )                    ⁢                          ⁢              ⅆ        τ            and the conservation variable at outlet of the pipe is defined as
                              L          o                ⁡                  (          t          )                    ⁢                        ∫                      t            0                    t                ⁢                                            v              o                        ⁡                          (              τ              )                                ⁢                                          ⁢                      ⅆ            τ                    ⁢                                                            ⁢                                                          ⁢          then          ⁢                                          ⁢                                    L              i                        ⁡                          (              t              )                                            -                  L        o            ⁡              (        t        )              =      L    .  Additionally, as Lo(t)=Li(t−td), it is therefore evident that:νi(t)=νo(t)Li(t)−Li(t−td)=L. Note, however, that the delay set forth above is only constant if the velocity is constant, making it only possible to study velocity related properties or the conservation variable at the inlet and the outlet using the above equation.
In many modern systems, however, such a constant velocity is not applicable. For example, fluid flow velocity through the aforementioned pipe may be increased by the addition of a variable speed pump, such that velocity of the fluid is no longer constant. In view of this, a need to study some other property associated with the system at the input and output of the system is required. Namely, a need to determine a variable transport delay wherein the variables are not bound by existing assumptions is required to more accurately model dynamic systems.